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International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume: 3 | Issue: 4 | May-Jun 2019 Available Online: www.ijtsrd.com e-ISSN: 2456 - 6470
@ IJTSRD | Unique Paper ID - IJTSRD23768 | Volume – 3 | Issue – 4 | May-Jun 2019 Page: 398
Implementation of Fractal Image Compression
on Medical Images by Different Approach
Mr. Vaibhav Vijay Bulkunde1, Mr. Nilesh P. Bodne1, Dr. Sunil Kumar2
1Student, 2Professor
1,2ECE Department, Kalinga University, Raipur, Chhattisgarh, India
How to cite this paper: Mr. Vaibhav
Vijay Bulkunde | Mr. Nilesh P. Bodne |
Dr. Sunil Kumar "Implementation of
Fractal Image Compression on Medical
Images by Different Approach"
Published in International Journal of
Trend in Scientific Research and
Development
(ijtsrd), ISSN: 2456-
6470, Volume-3 |
Issue-4, June 2019,
pp.398-400, URL:
https://www.ijtsrd.c
om/papers/ijtsrd23
768.pdf
Copyright © 2019 by author(s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
(http://creativecommons.org/licenses/
by/4.0)
ABSTRACT
FIC (Fractal Image Processing) is actually a JPGimagewhichneedstobeperform
large scale encoding to improveand increasethe compression ratio.Inthispaper
we are going to analyze different constraints and algorithmsof imageprocessing
in MATLAB so that there will be very low loss in image quality. It has been seen
that whenever we contain any HD picture that to in medical applicationweneed
to sharpen the image that is we need to perform image encryption, image de-
noising and there should be no loss in image quality. In this paper, we actualized
both consecutive and parallel adaptation of fractal picture pressure calculations
utilizing programmingshow for parallelizingtheprogram in GraphicsProcessing
Unit for medicinal pictures, as they are very comparable inside the pictureitself.
Whenever we consider an image into fractal image, it has great importance and
application in image processing field. In this paper compression scheme is used
to sharpen and smoothen of image by using various imageprocessingalgorithm.
There are a few enhancements in the usage of the calculation too.Fractalpicture
pressure is based on the self-closeness of a picture, which means a picture
having closeness in dominant part of the locales. We accept this open door to
execute the pressure calculation and screen the impact of it utilizing both
parallel and successive execution. Fractal pressure has the property of high
pressure rate and the dimensionless plan. Pressure plot for fractal picture is of
two kind, one is encoding and another is deciphering. Encoding is especially
computational costly. Then again interpreting is less computational. The use of
fractal pressure to restorative pictures would permit acquiring a lot higher
pressure proportions. While the fractal amplification an indivisible element of
the fractal pressure would be helpful in showing the recreated picture in an
exceedingly meaningful structure. Be that as it may, similar to all irreversible
strategies, the fractal pressure is associated with the issue of data misfortune,
which is particularly troublesome in the therapeutic imaging. A very tedious
encoding process, which can last even a few hours, is another troublesome
downside of the fractal pressure.
Keywords: FIC (Fractal Image Processing), sharpening, smoothing, image
compression
INTRODUCTION
Fractal Image Compression (FIC) is actually an ideainwhich
the image is divided into sub-blocks, to do fractal
compression. A resolution independent of any image of size
128×128 has to be analyze. The coding, storage and
reconstruction of images remains an important factor in the
application of computer technique to technical andscientific
problem. Our research has focus mainly on thedevelopment
of fractal method forcompressingimagedata. Thismethodis
loss free and hence high compression ratio. Our method are
computationally intensive but are feasible for non-real time
application on computers.Theimage compressiontechnique
which we are going to use in this paper has purpose of
increasing compression rate and computational time.
Research on image processing present number of
challenging field. One amongstthemisimage compression. A
common characteristic of most images is that theyareprone
to computer redundancy. A number of diverse compression
technique from lossy compressiontolosslesscompression is
encountered.
Fractal Image Compression (FIC) is relatively recent image
compression method which are used for different part of
image. The FIC algorithm initially begins with the complete
image and partition the imageintonumberof smallerblocks.
Fractal compression uses encoding and decoding technique
also. As we know enormous amount of therapeutic pictures
are created regular and those are to be put away in some
spot for future reference and utilization. It requires a great
deal stockpiling and investment to store them. In this
manner we demonstrate that on the off chance that we use
GPUs to process the picture what's more, diminish the size
for putting away, it bodes well. There may a few inquiries
emerge that for what reason to consider restorativepictures
as it were. The fundamental explanation for this ismedicinal
IJTSRD23768
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID - IJTSRD23768 | Volume – 3 | Issue – 4 | May-Jun 2019 Page: 399
pictures are self-comparative furthermore, can be
compacted effectively with better size decrease [18]. It is
portrayed in more detail in the later segments. Then again,
pictures with the exception of therapeutic zones, thegreater
part of them are not highly contrasting. They are mind
boggling in nature which means the picture contain more
data contrasted with highly contrasting. Along these lines it
isn't useful to pack this kind of pictures. We won't get much
decrease of the size after pressure as we have to put a large
portion of the data flawless to get back the first picture [19].
Fractal picture pressure [2] technique will encode a picture
as a lot of changes that are extremely comparable to the
duplicate machine plot. For instance, thebranchhasdetailat
each level, the figure remade from the changes as well. The
picture can be decoded at any size since it has no typical
range. The extra component required for unraveling at
bigger scale is generated involuntarily by the encoding
transforms. We could decode an image of an individual
escalating the size with each iteration, and ultimately can
observe skin cells or perhaps atoms; it may be surprising if
this detail is real. The response is in fact no. At what time the
image was digitized, the detail is not at all associated to the
actual detail present; it is just the artifact of the encoding
transforms which initially only encoded the large- scale
features. Nonetheless, in some occasion the detail is
pragmatic at low magnifications, and this can be handy in
Security and Medical imaging applications. As we know that
the compression is based on findingthesimilarfragmentand
encoding them accordingly, thus there is redundant
processing involved. So, the processing time becomes less if
these redundant processing can be deployed to number of
processors. This can be done with the help of parallel
graphics processors. Hence, CUDA comes in the way to
perform parallel processing. The remainder of this paper is
organized as follows. In section II, we mentioned a concise
overview of the related work done in this arena. We
explained fractal image compression briefly in section III. In
section IV, mentioning how we prepared our images, CPU
and GPU configuration. In section V, we demonstrate.
FRACTAL IMAGE COMPRESSION (FIC)
A new compression technique recently developed to
compress the image is Fractal Image Compression (FIC). In
the recent years FIC in many applications is utilize not only
in medical but also in remote sensing geographic scheme.
Different algorithms has been proposed to implement
Fractal Image Compression based on partitional function.In
other words we can say that we have to calculate the matrix
of any image in the metrics form and then we have to apply
image processing algorithms in wavelet transform in image
processing. Each algorithm follows the same step. The
classification of the system is carried out with the help of
wavelet 3D transform in image processing using MATLAB.
PERFORMANCE ANALYSIS
In this section we will aquise that things in which image
compression technique is well suited within compression
time and hence high performance in reduction of
computational time. The FIC uses a gray level scale image of
Lena of size 128×128 has been considered for training the
network using back propagation algorithm.A domainpoolis
created of size 4×4 for the Lena image and the standard
deviation with histogram for Lena image has been carried
out with special collaboration classes. The computer
simulation has been carried outonMATLABenvironmenton
Intel® Core™ i5-2430M CPU @ 2.40 GHz processor of
4.00GB installed memory(RAM).
The image compression technique used in this project is
found to have low Signal to Noise (SNR) with high encoding
time and low decoding time. This algorithm is wellsuited for
any color image also of size 256×256.
METHODOLOGY
Two classes of digital images can be distinguished – analog
and digital images. Both types fall into non-temporal
multimedia type. Analog images are painted or created
through photographic process. During this process, the
image is captured by a camera on a film that becomes a
negative. We have a positive when the film is developed– no
processing is possible from this moment. When the
photography is made on a transparent medium then we are
dealing with a diapositive (slide). Analog images are
characterizedbycontinuous,smooth transitionof tones.This
means that between each two different points at the picture
there is an infinite number of tonal values. It is possible to
transform an analog image into digital. The digitization
process is usually caused by a need of digital processing.The
output of digitalization isadigitalapproximation of theinput
analog image – the analog image is replaced by a setofpixels
(points organized in rows and columns) and every pixel has
a fixed, discrete tone value. Therefore, the image is not a
continuous tone of colors. The precision and accuracyof this
transformation depends on the size of a pixel – the larger
area of an analog image transformed into one pixel the less
precise approximation.
A. The key issues in FIC are properselectionofpartitioning
schemes to partition the image into domain and range
blocks, transformation techniques to map the domain
blocks to range blocks and speed up approaches to
lessen the encoding time.
B. The present work while addressing the above problem
has developed some approaches to enhance the
performance of FIC on medical images. In the current
work initially feasibility/suitability of FIC to medical
images is investigated usingstandardconventionalfixed
partitioning FIC algorithm and further some speed up
approaches is applied on medical images to reduce the
encoding time.
C. The novel architecture for FIC using quad tree based
partitioning technique for medical images has been
implemented, modeled using MATLAB
Fig: 1 Block Diagram of FIC
CONCLUSION
In this project we have done successful analysis of FIC
method of medical sciences and the performances has been
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID - IJTSRD23768 | Volume – 3 | Issue – 4 | May-Jun 2019 Page: 400
carried out. It is found that this technique is governed with
single attributes and found out to be very efficient, lossless
and high compression and low functionality of
computational time.
Fig: 2 Output before and after compression of image of a
skull
Fig: 3 Output before and after compression of image of a X-
ray of Leg
REFRENCES
[1] Mielikainen, Jarno, et al. Constant coefficients linear
prediction for lossless compression of ultra-spectral
sounder data using a graphics processing unit. Journal
of Applied Remote Sensing 4.1 (2010): 041774-
041774.
[2] M.F.Barnsley, L.P.Hurd andL.F.Anson,Fractal image
compression. AK peters Wellesley, 1993.
[3] Wakatani, Akiyoshi. Improvement of adaptive fractal
image coding on GPUs. Consumer Electronics (ICCE),
2012 IEEE International Conference on. IEEE, 2012.
[4] Franco, Joaqun, et al. A parallel implementation of the
2D wavelet transformusingCUDA.Parallel,Distributed
and Network-based Processing, 2009 17th Euromicro
International Conference on. IEEE, 2009.
[5] Simek, Vaclav, and Ram Rakesh Asn. ”GPU acceleration
of 2d-dwt image compression in matlab with cuda.”
Computer Modeling and Simulation, 2008. EMS’08.
Second UKSIM European Symposium on. IEEE, 2008.
[6] Lin, Caiwei, Lei Zhao, and Jiwen Yang. A High
Performance Image Authentication Algorithm on GPU
with CUDA. International Journalof IntelligentSystems
and Applications (IJISA) 3.2 (2011): 52.
[7] Park, In Kyu, et al. Design and performance evaluation
of image processing algorithms on GPUs. Parallel and
Distributed Systems, IEEE Transactions on 22.1
(2011): 91-104.
[8] Monro, Donald M., and Frank Dudbridge. Fractal
approximation of image blocks. Acoustics, Speech, and
Signal Processing, 1992. ICASSP-92., 1992 IEEE
International Conference on. Vol. 3. IEEE, 1992.
[9] Hutchinson, John E., and Ludger Rschendorf. Random
fractal measures via the contractionmethod.Centrefor
Mathematics and its Applications, School of
Mathematical Sciences, Australian NationalUniversity,
1997.
[10] Marquardt, Donald W. An algorithm for least-squares
estimation of nonlinear parameters. Journal of the
Society for Industrial & Applied Mathematics 11.2
(1963): 431-441.
[11] Oien, G. E., et al. A new improved collage theorem with
applications to multi resolution fractal image coding.
Acoustics,Speech, andSignal Processing, 1994.ICASSP-
94., 1994 IEEE International Conference on. IEEE,
1994.
[12] Falconer, Kenneth. Fractal geometry: mathematical
foundations and applications. Wiley.com, 2007.
[13] Mandelbrot, Benoit B., and A. Blumen. Fractal
Geometry: What is it, and What Does it do? [and
Discussion]. Proceedings of the Royal Society of
London. A. Mathematical and Physical Sciences
423.1864 (1989): 3- 16.
[14] Dasgupta, Abhijit. Brouwers Theorem and Sierpinskis
Theorem. Set Theory. Springer New York, 2014. 313-
319.
[15] Barnsley, Michael F., et al. Solution of an inverse
problem for fractals and other sets. Proceedings of the
National Academy of Sciences of the United States of
America 83.7 (1986): 1975.
[16] Erra, Ugo. Toward real time fractal image compression
using graphics hardware. Advances in Visual
Computing. SpringerBerlin Heidelberg,2005.723-728.
[17] Sodora, August. Fractal Image Compression on the
Graphics Card. Hopkins Undergraduate Research
Journal, Johns Hopkins University (2010).
[18] Che, Shuai, et al. A performance study of general-
purpose applications on graphics processors using
CUDA. Journal of parallel and distributed computing
68.10 (2008): 1370-1380.

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Implementation of Fractal Image Compression on Medical Images by Different Approach

  • 1. International Journal of Trend in Scientific Research and Development (IJTSRD) Volume: 3 | Issue: 4 | May-Jun 2019 Available Online: www.ijtsrd.com e-ISSN: 2456 - 6470 @ IJTSRD | Unique Paper ID - IJTSRD23768 | Volume – 3 | Issue – 4 | May-Jun 2019 Page: 398 Implementation of Fractal Image Compression on Medical Images by Different Approach Mr. Vaibhav Vijay Bulkunde1, Mr. Nilesh P. Bodne1, Dr. Sunil Kumar2 1Student, 2Professor 1,2ECE Department, Kalinga University, Raipur, Chhattisgarh, India How to cite this paper: Mr. Vaibhav Vijay Bulkunde | Mr. Nilesh P. Bodne | Dr. Sunil Kumar "Implementation of Fractal Image Compression on Medical Images by Different Approach" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456- 6470, Volume-3 | Issue-4, June 2019, pp.398-400, URL: https://www.ijtsrd.c om/papers/ijtsrd23 768.pdf Copyright © 2019 by author(s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/ by/4.0) ABSTRACT FIC (Fractal Image Processing) is actually a JPGimagewhichneedstobeperform large scale encoding to improveand increasethe compression ratio.Inthispaper we are going to analyze different constraints and algorithmsof imageprocessing in MATLAB so that there will be very low loss in image quality. It has been seen that whenever we contain any HD picture that to in medical applicationweneed to sharpen the image that is we need to perform image encryption, image de- noising and there should be no loss in image quality. In this paper, we actualized both consecutive and parallel adaptation of fractal picture pressure calculations utilizing programmingshow for parallelizingtheprogram in GraphicsProcessing Unit for medicinal pictures, as they are very comparable inside the pictureitself. Whenever we consider an image into fractal image, it has great importance and application in image processing field. In this paper compression scheme is used to sharpen and smoothen of image by using various imageprocessingalgorithm. There are a few enhancements in the usage of the calculation too.Fractalpicture pressure is based on the self-closeness of a picture, which means a picture having closeness in dominant part of the locales. We accept this open door to execute the pressure calculation and screen the impact of it utilizing both parallel and successive execution. Fractal pressure has the property of high pressure rate and the dimensionless plan. Pressure plot for fractal picture is of two kind, one is encoding and another is deciphering. Encoding is especially computational costly. Then again interpreting is less computational. The use of fractal pressure to restorative pictures would permit acquiring a lot higher pressure proportions. While the fractal amplification an indivisible element of the fractal pressure would be helpful in showing the recreated picture in an exceedingly meaningful structure. Be that as it may, similar to all irreversible strategies, the fractal pressure is associated with the issue of data misfortune, which is particularly troublesome in the therapeutic imaging. A very tedious encoding process, which can last even a few hours, is another troublesome downside of the fractal pressure. Keywords: FIC (Fractal Image Processing), sharpening, smoothing, image compression INTRODUCTION Fractal Image Compression (FIC) is actually an ideainwhich the image is divided into sub-blocks, to do fractal compression. A resolution independent of any image of size 128×128 has to be analyze. The coding, storage and reconstruction of images remains an important factor in the application of computer technique to technical andscientific problem. Our research has focus mainly on thedevelopment of fractal method forcompressingimagedata. Thismethodis loss free and hence high compression ratio. Our method are computationally intensive but are feasible for non-real time application on computers.Theimage compressiontechnique which we are going to use in this paper has purpose of increasing compression rate and computational time. Research on image processing present number of challenging field. One amongstthemisimage compression. A common characteristic of most images is that theyareprone to computer redundancy. A number of diverse compression technique from lossy compressiontolosslesscompression is encountered. Fractal Image Compression (FIC) is relatively recent image compression method which are used for different part of image. The FIC algorithm initially begins with the complete image and partition the imageintonumberof smallerblocks. Fractal compression uses encoding and decoding technique also. As we know enormous amount of therapeutic pictures are created regular and those are to be put away in some spot for future reference and utilization. It requires a great deal stockpiling and investment to store them. In this manner we demonstrate that on the off chance that we use GPUs to process the picture what's more, diminish the size for putting away, it bodes well. There may a few inquiries emerge that for what reason to consider restorativepictures as it were. The fundamental explanation for this ismedicinal IJTSRD23768
  • 2. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID - IJTSRD23768 | Volume – 3 | Issue – 4 | May-Jun 2019 Page: 399 pictures are self-comparative furthermore, can be compacted effectively with better size decrease [18]. It is portrayed in more detail in the later segments. Then again, pictures with the exception of therapeutic zones, thegreater part of them are not highly contrasting. They are mind boggling in nature which means the picture contain more data contrasted with highly contrasting. Along these lines it isn't useful to pack this kind of pictures. We won't get much decrease of the size after pressure as we have to put a large portion of the data flawless to get back the first picture [19]. Fractal picture pressure [2] technique will encode a picture as a lot of changes that are extremely comparable to the duplicate machine plot. For instance, thebranchhasdetailat each level, the figure remade from the changes as well. The picture can be decoded at any size since it has no typical range. The extra component required for unraveling at bigger scale is generated involuntarily by the encoding transforms. We could decode an image of an individual escalating the size with each iteration, and ultimately can observe skin cells or perhaps atoms; it may be surprising if this detail is real. The response is in fact no. At what time the image was digitized, the detail is not at all associated to the actual detail present; it is just the artifact of the encoding transforms which initially only encoded the large- scale features. Nonetheless, in some occasion the detail is pragmatic at low magnifications, and this can be handy in Security and Medical imaging applications. As we know that the compression is based on findingthesimilarfragmentand encoding them accordingly, thus there is redundant processing involved. So, the processing time becomes less if these redundant processing can be deployed to number of processors. This can be done with the help of parallel graphics processors. Hence, CUDA comes in the way to perform parallel processing. The remainder of this paper is organized as follows. In section II, we mentioned a concise overview of the related work done in this arena. We explained fractal image compression briefly in section III. In section IV, mentioning how we prepared our images, CPU and GPU configuration. In section V, we demonstrate. FRACTAL IMAGE COMPRESSION (FIC) A new compression technique recently developed to compress the image is Fractal Image Compression (FIC). In the recent years FIC in many applications is utilize not only in medical but also in remote sensing geographic scheme. Different algorithms has been proposed to implement Fractal Image Compression based on partitional function.In other words we can say that we have to calculate the matrix of any image in the metrics form and then we have to apply image processing algorithms in wavelet transform in image processing. Each algorithm follows the same step. The classification of the system is carried out with the help of wavelet 3D transform in image processing using MATLAB. PERFORMANCE ANALYSIS In this section we will aquise that things in which image compression technique is well suited within compression time and hence high performance in reduction of computational time. The FIC uses a gray level scale image of Lena of size 128×128 has been considered for training the network using back propagation algorithm.A domainpoolis created of size 4×4 for the Lena image and the standard deviation with histogram for Lena image has been carried out with special collaboration classes. The computer simulation has been carried outonMATLABenvironmenton Intel® Core™ i5-2430M CPU @ 2.40 GHz processor of 4.00GB installed memory(RAM). The image compression technique used in this project is found to have low Signal to Noise (SNR) with high encoding time and low decoding time. This algorithm is wellsuited for any color image also of size 256×256. METHODOLOGY Two classes of digital images can be distinguished – analog and digital images. Both types fall into non-temporal multimedia type. Analog images are painted or created through photographic process. During this process, the image is captured by a camera on a film that becomes a negative. We have a positive when the film is developed– no processing is possible from this moment. When the photography is made on a transparent medium then we are dealing with a diapositive (slide). Analog images are characterizedbycontinuous,smooth transitionof tones.This means that between each two different points at the picture there is an infinite number of tonal values. It is possible to transform an analog image into digital. The digitization process is usually caused by a need of digital processing.The output of digitalization isadigitalapproximation of theinput analog image – the analog image is replaced by a setofpixels (points organized in rows and columns) and every pixel has a fixed, discrete tone value. Therefore, the image is not a continuous tone of colors. The precision and accuracyof this transformation depends on the size of a pixel – the larger area of an analog image transformed into one pixel the less precise approximation. A. The key issues in FIC are properselectionofpartitioning schemes to partition the image into domain and range blocks, transformation techniques to map the domain blocks to range blocks and speed up approaches to lessen the encoding time. B. The present work while addressing the above problem has developed some approaches to enhance the performance of FIC on medical images. In the current work initially feasibility/suitability of FIC to medical images is investigated usingstandardconventionalfixed partitioning FIC algorithm and further some speed up approaches is applied on medical images to reduce the encoding time. C. The novel architecture for FIC using quad tree based partitioning technique for medical images has been implemented, modeled using MATLAB Fig: 1 Block Diagram of FIC CONCLUSION In this project we have done successful analysis of FIC method of medical sciences and the performances has been
  • 3. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID - IJTSRD23768 | Volume – 3 | Issue – 4 | May-Jun 2019 Page: 400 carried out. It is found that this technique is governed with single attributes and found out to be very efficient, lossless and high compression and low functionality of computational time. Fig: 2 Output before and after compression of image of a skull Fig: 3 Output before and after compression of image of a X- ray of Leg REFRENCES [1] Mielikainen, Jarno, et al. Constant coefficients linear prediction for lossless compression of ultra-spectral sounder data using a graphics processing unit. Journal of Applied Remote Sensing 4.1 (2010): 041774- 041774. [2] M.F.Barnsley, L.P.Hurd andL.F.Anson,Fractal image compression. AK peters Wellesley, 1993. [3] Wakatani, Akiyoshi. Improvement of adaptive fractal image coding on GPUs. Consumer Electronics (ICCE), 2012 IEEE International Conference on. IEEE, 2012. [4] Franco, Joaqun, et al. A parallel implementation of the 2D wavelet transformusingCUDA.Parallel,Distributed and Network-based Processing, 2009 17th Euromicro International Conference on. IEEE, 2009. [5] Simek, Vaclav, and Ram Rakesh Asn. ”GPU acceleration of 2d-dwt image compression in matlab with cuda.” Computer Modeling and Simulation, 2008. EMS’08. Second UKSIM European Symposium on. IEEE, 2008. [6] Lin, Caiwei, Lei Zhao, and Jiwen Yang. A High Performance Image Authentication Algorithm on GPU with CUDA. International Journalof IntelligentSystems and Applications (IJISA) 3.2 (2011): 52. [7] Park, In Kyu, et al. Design and performance evaluation of image processing algorithms on GPUs. Parallel and Distributed Systems, IEEE Transactions on 22.1 (2011): 91-104. [8] Monro, Donald M., and Frank Dudbridge. Fractal approximation of image blocks. Acoustics, Speech, and Signal Processing, 1992. ICASSP-92., 1992 IEEE International Conference on. Vol. 3. IEEE, 1992. [9] Hutchinson, John E., and Ludger Rschendorf. Random fractal measures via the contractionmethod.Centrefor Mathematics and its Applications, School of Mathematical Sciences, Australian NationalUniversity, 1997. [10] Marquardt, Donald W. An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial & Applied Mathematics 11.2 (1963): 431-441. [11] Oien, G. E., et al. A new improved collage theorem with applications to multi resolution fractal image coding. Acoustics,Speech, andSignal Processing, 1994.ICASSP- 94., 1994 IEEE International Conference on. IEEE, 1994. [12] Falconer, Kenneth. Fractal geometry: mathematical foundations and applications. Wiley.com, 2007. [13] Mandelbrot, Benoit B., and A. Blumen. Fractal Geometry: What is it, and What Does it do? [and Discussion]. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 423.1864 (1989): 3- 16. [14] Dasgupta, Abhijit. Brouwers Theorem and Sierpinskis Theorem. Set Theory. Springer New York, 2014. 313- 319. [15] Barnsley, Michael F., et al. Solution of an inverse problem for fractals and other sets. Proceedings of the National Academy of Sciences of the United States of America 83.7 (1986): 1975. [16] Erra, Ugo. Toward real time fractal image compression using graphics hardware. Advances in Visual Computing. SpringerBerlin Heidelberg,2005.723-728. [17] Sodora, August. Fractal Image Compression on the Graphics Card. Hopkins Undergraduate Research Journal, Johns Hopkins University (2010). [18] Che, Shuai, et al. A performance study of general- purpose applications on graphics processors using CUDA. Journal of parallel and distributed computing 68.10 (2008): 1370-1380.